Bifurcation Analysis and Control of Leslie–Gower Predator–Prey Model with Michaelis–Menten Type Prey-Harvesting

Abstract

In this paper, a bi-dimensional system of ordinary differential equation prey–predator model with nonlinear prey-harvesting is analyzed. The model is a modified version of the well known Leslie–Gower type prey–predator model. The original Leslie–Gower type model has the unique interior equilibrium point which is globally asymptotically stable. We show here that the nonlinear harvesting in prey significantly modifies the dynamics of the system in comparison to the proportionate harvesting of prey. It has been observed that the system goes to extinction for a wide range of initial values. Moreover, the model can have two, one or no interior equilibrium point in the first quadrant where two interior equilibria collapse to one interior equilibrium point and then disappear through saddle-node bifurcation considering the rate of harvesting as bifurcation parameter. The local existence of limit cycle appearing through local Hopf bifurcation and its stability have also been investigated by computing first Lyapunov number. The conditions for existence of bionomic equilibria are analyzed. Pontryagin’s Maximum Principle is used to characterize the optimal singular control. Numerical examples are provided to validate our findings.